Hokkaido Mathematical Journal

Spectral gaps of the one-dimensional Schr\"odinger operators with periodic point interactions

Kazushi YOSHITOMI

Full-text: Open access

Abstract

We study the spectral gaps of the Schr{\"o}dinger operators $$H_{1}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta^{\prime}(x-\kappa-2\pi l)+\beta_{2}\delta^{\prime}(x-2\pi l))\quad {\rm in}\quad L^{2}({\mathbb R}),$$ $$H_{2}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta(x-\kappa-2\pi l)+\beta_{2}\delta(x-2\pi l))\quad {\rm in}\quad L^{2}({\mathbb R}),$$ where $\kappa\in (0,2\pi)$ and $\beta_{1},\beta_{2}\in{\mathbb R}\backslash\{0\}$ are parameters. Given $j\in{\mathbb N}$, we determine whether the $j$th gap of $H_{k}$ is absent or not for $k=1,2$.

Article information

Source
Hokkaido Math. J., Volume 35, Number 2 (2006), 365-378.

Dates
First available in Project Euclid: 29 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1285766361

Digital Object Identifier
doi:10.14492/hokmj/1285766361

Mathematical Reviews number (MathSciNet)
MR2254656

Zentralblatt MATH identifier
1114.34067

Subjects
Primary: 34B37: Boundary value problems with impulses
Secondary: 34D08: Characteristic and Lyapunov exponents 34B30: Special equations (Mathieu, Hill, Bessel, etc.) 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)

Keywords
Schr\"odinger operators periodic point interactions spectral gaps

Citation

YOSHITOMI, Kazushi. Spectral gaps of the one-dimensional Schr\"odinger operators with periodic point interactions. Hokkaido Math. J. 35 (2006), no. 2, 365--378. doi:10.14492/hokmj/1285766361. https://projecteuclid.org/euclid.hokmj/1285766361


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